Video solution:

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Written solution:

Place $A=(0,0,0)$, $B=(2,0,0)$, $C=(0,3,0)$, and $D=(0,0,4)$. Then

$$BC=\sqrt{2^2+3^2}=\sqrt{13},\quad BD=\sqrt{2^2+4^2}=2\sqrt5,\quad CD=\sqrt{3^2+4^2}=5,$$

so this coordinate model matches all the given edge lengths. The tetrahedron is a rectangular-corner tetrahedron with perpendicular edge lengths $2,3,4$ from $A$, so its volume is

$$\frac16(2)(3)(4)=4.$$

Thus, C is the correct answer.